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Named after its inventor, E.Ch. Catalan (1814–1894), the Catalan constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c1300401.png" /> (which is denoted also by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c1300402.png" />) is defined by
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c1300403.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c1300404.png" /></td> </tr></table>
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Named after its inventor, E.Ch. Catalan (1814–1894), the Catalan constant $G$ (which is denoted also by $\lambda$) is defined by
  
If, in terms of the Digamma (or Psi) function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c1300405.png" />, defined by
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\begin{equation} \label{a1} G : = \sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { ( 2 k + 1 ) ^ { 2 } } \approx0.915965594177219015 \ldots. \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c1300406.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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If, in terms of the Digamma (or Psi) function $\psi ( z )$, defined by
 +
 
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\begin{equation} \label{a2} \psi ( z ) : = \frac { d } { d z } \{ \operatorname { log } \Gamma ( z ) \} = \frac { \Gamma ^ { \prime } ( z ) } { \Gamma ( z ) } \end{equation}
  
 
or
 
or
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c1300407.png" /></td> </tr></table>
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\begin{equation*} \operatorname { log } \Gamma ( z ) = \int _ { 1 } ^ { z } \psi ( t ) d t, \end{equation*}
  
 
one puts
 
one puts
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c1300408.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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\begin{equation} \label{a3} \beta ( z ) : = \frac { 1 } { 2 } \left[ \psi \left( \frac { 1 } { 2 } z + \frac { 1 } { 2 } \right) - \psi \left( \frac { 1 } { 2 } z \right) \right] = \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c1300409.png" /></td> </tr></table>
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\begin{equation*} = \sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { z + k }, \end{equation*}
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004010.png" /></td> </tr></table>
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\begin{equation*} z \in \mathbf{C} \backslash \mathbf{Z} _ { 0 }^- , \quad \mathbf{Z} _ { 0 } ^ { - } : = \{ 0 , - 1 , - 2 , \ldots \}, \end{equation*}
  
 
then
 
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
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\begin{equation} \label{a4} G = - \frac { 1 } { 4 } \beta ^ { \prime } \left( \frac { 1 } { 2 } \right) \end{equation}
  
which provides a relationship between the Catalan constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004012.png" /> and the Digamma function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004013.png" />.
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which provides a relationship between the Catalan constant $G$ and the Digamma function $\psi ( z )$.
  
The Catalan constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004014.png" /> is related also to other functions, such as the Clausen function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004015.png" />, defined by
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The Catalan constant $G$ is related also to other functions, such as the Clausen function $\operatorname{Cl}_{2} (z)$, defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
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\begin{equation} \label{a5}  \operatorname {Cl} _ { 2 } ( z ) : = - \int _ { 0 } ^ { z } \operatorname { log } \left| 2 \operatorname { sin } \left( \frac { 1 } { 2 } t \right) \right| d t = \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004017.png" /></td> </tr></table>
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\begin{equation*} = \sum _ { k = 1 } ^ { \infty } \frac { \operatorname { sin } ( k z ) } { k ^ { 2 } }, \end{equation*}
  
and the [[Hurwitz zeta function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004018.png" />, which is defined, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004019.png" />, by
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and the [[Hurwitz zeta function]] $\zeta ( s , a )$, which is defined, when $\operatorname { Re } s > 1$, by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
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\begin{equation} \label{a6} \zeta ( s , a ) : = \sum _ { k = 0 } ^ { \infty } \frac { 1 } { ( k + a ) ^ { s } }, \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004021.png" /></td> </tr></table>
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\begin{equation*} \operatorname { Re } s > 1 , a \in \mathbf{C} \backslash \mathbf{Z} ^{ - } _ { 0 }. \end{equation*}
  
 
Thus,
 
Thus,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
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\begin{equation} \label{a7} G = \operatorname{Cl} _ { 2 } ( \frac { 1 } { 2 } \pi ) = - \operatorname{Cl} _ { 2 } \left( \frac { 3 } { 2 } \pi \right) = \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004023.png" /></td> </tr></table>
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\begin{equation*} = \frac { 1 } { 16 } \left[ \zeta \left( 2 , \frac { 1 } { 4 } \right) - \zeta \left( 2 , \frac { 3 } { 4 } \right) \right]. \end{equation*}
  
 
Since
 
Since
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>
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\begin{equation} \label{a8} \psi ^ { ( n ) } ( z ) = ( - 1 ) ^ { n + 1 } n ! \zeta ( n + 1 , z ), \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004025.png" /></td> </tr></table>
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\begin{equation*} n \in \mathbf{N} : = \{ 1,2 , \ldots \} , z \in \mathbf{C} \backslash \mathbf{Z} _ { 0 } ^ { - }, \end{equation*}
  
the last expression in (a7) would follow also from (a4) in light of the definition in (a3).
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the last expression in \eqref{a7} would follow also from \eqref{a4} in light of the definition in \eqref{a3}.
  
A fairly large number of integrals and series can be evaluated in terms of the Catalan constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004026.png" />. For example,
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A fairly large number of integrals and series can be evaluated in terms of the Catalan constant $G$. For example,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>
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\begin{equation} \label{a9} \int _ { 0 } ^ { 1 } \frac { t\operatorname { log } ( t ^ { - 1 } \pm t ) } { 1 + t ^ { 4 } } d t = \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004028.png" /></td> </tr></table>
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\begin{equation*} = \int _ { 1 } ^ { \infty } \frac { t \operatorname { log } ( t \pm t ^ { - 1 } ) } { 1 + t ^ { 4 } } d t = \frac { \pi } { 16 } \operatorname { log } 2 \pm \frac { G } { 4 }, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a10)</td></tr></table>
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\begin{equation} \label{a10} \sum _ { k = 1 } ^ { \infty } \left( \frac { ( 2 k + 1 ) ! } { k ! ( k + 1 ) ! } \right) ^ { 2 } \frac { 2 ^ { - 4 k } } { k } = \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004030.png" /></td> </tr></table>
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\begin{equation*} = 4 \operatorname { log } 2 + 2 - \frac { 4 } { \pi } ( 2 G + 1 ), \end{equation*}
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a11)</td></tr></table>
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\begin{equation} \label{a11} \sum _ { k = 1 } ^ { \infty } \frac { \zeta ( 2 k ) } { k ( 2 k + 1 ) 2 ^ { 4 k } } = \operatorname { log } ( \frac { \pi } { 2 } ) - 1 + \frac { 2 G } { \pi }, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004032.png" /> denotes the familiar [[Riemann zeta-function|Riemann zeta-function]].
+
where $\zeta ( s ) = \zeta ( s , 1 )$ denotes the familiar [[Riemann zeta-function|Riemann zeta-function]].
  
 
====References====
 
====References====
{|
+
* {{Ref|Fi}} Steven R. Finch, "Mathematical constants", Encyclopedia of mathematics and its applications '''94''', Cambridge University Press (2003) {{ISBN|0-521-81805-2}} {{ZBL|1054.00001}}
|-
 
|valign="top"|{{Ref|Fi}}||valign="top"|  Steven R. Finch,   "Mathematical constants" , Encyclopedia of mathematics and its applications '''94''', Cambridge University Press (2003) ISBN 0-521-81805-2 {{ZBL|1054.00001}}
 
|-
 
|}
 

Latest revision as of 14:03, 11 November 2023

Named after its inventor, E.Ch. Catalan (1814–1894), the Catalan constant $G$ (which is denoted also by $\lambda$) is defined by

\begin{equation} \label{a1} G : = \sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { ( 2 k + 1 ) ^ { 2 } } \approx0.915965594177219015 \ldots. \end{equation}

If, in terms of the Digamma (or Psi) function $\psi ( z )$, defined by

\begin{equation} \label{a2} \psi ( z ) : = \frac { d } { d z } \{ \operatorname { log } \Gamma ( z ) \} = \frac { \Gamma ^ { \prime } ( z ) } { \Gamma ( z ) } \end{equation}

or

\begin{equation*} \operatorname { log } \Gamma ( z ) = \int _ { 1 } ^ { z } \psi ( t ) d t, \end{equation*}

one puts

\begin{equation} \label{a3} \beta ( z ) : = \frac { 1 } { 2 } \left[ \psi \left( \frac { 1 } { 2 } z + \frac { 1 } { 2 } \right) - \psi \left( \frac { 1 } { 2 } z \right) \right] = \end{equation}

\begin{equation*} = \sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { z + k }, \end{equation*}

where

\begin{equation*} z \in \mathbf{C} \backslash \mathbf{Z} _ { 0 }^- , \quad \mathbf{Z} _ { 0 } ^ { - } : = \{ 0 , - 1 , - 2 , \ldots \}, \end{equation*}

then

\begin{equation} \label{a4} G = - \frac { 1 } { 4 } \beta ^ { \prime } \left( \frac { 1 } { 2 } \right) \end{equation}

which provides a relationship between the Catalan constant $G$ and the Digamma function $\psi ( z )$.

The Catalan constant $G$ is related also to other functions, such as the Clausen function $\operatorname{Cl}_{2} (z)$, defined by

\begin{equation} \label{a5} \operatorname {Cl} _ { 2 } ( z ) : = - \int _ { 0 } ^ { z } \operatorname { log } \left| 2 \operatorname { sin } \left( \frac { 1 } { 2 } t \right) \right| d t = \end{equation}

\begin{equation*} = \sum _ { k = 1 } ^ { \infty } \frac { \operatorname { sin } ( k z ) } { k ^ { 2 } }, \end{equation*}

and the Hurwitz zeta function $\zeta ( s , a )$, which is defined, when $\operatorname { Re } s > 1$, by

\begin{equation} \label{a6} \zeta ( s , a ) : = \sum _ { k = 0 } ^ { \infty } \frac { 1 } { ( k + a ) ^ { s } }, \end{equation}

\begin{equation*} \operatorname { Re } s > 1 , a \in \mathbf{C} \backslash \mathbf{Z} ^{ - } _ { 0 }. \end{equation*}

Thus,

\begin{equation} \label{a7} G = \operatorname{Cl} _ { 2 } ( \frac { 1 } { 2 } \pi ) = - \operatorname{Cl} _ { 2 } \left( \frac { 3 } { 2 } \pi \right) = \end{equation}

\begin{equation*} = \frac { 1 } { 16 } \left[ \zeta \left( 2 , \frac { 1 } { 4 } \right) - \zeta \left( 2 , \frac { 3 } { 4 } \right) \right]. \end{equation*}

Since

\begin{equation} \label{a8} \psi ^ { ( n ) } ( z ) = ( - 1 ) ^ { n + 1 } n ! \zeta ( n + 1 , z ), \end{equation}

\begin{equation*} n \in \mathbf{N} : = \{ 1,2 , \ldots \} , z \in \mathbf{C} \backslash \mathbf{Z} _ { 0 } ^ { - }, \end{equation*}

the last expression in \eqref{a7} would follow also from \eqref{a4} in light of the definition in \eqref{a3}.

A fairly large number of integrals and series can be evaluated in terms of the Catalan constant $G$. For example,

\begin{equation} \label{a9} \int _ { 0 } ^ { 1 } \frac { t\operatorname { log } ( t ^ { - 1 } \pm t ) } { 1 + t ^ { 4 } } d t = \end{equation}

\begin{equation*} = \int _ { 1 } ^ { \infty } \frac { t \operatorname { log } ( t \pm t ^ { - 1 } ) } { 1 + t ^ { 4 } } d t = \frac { \pi } { 16 } \operatorname { log } 2 \pm \frac { G } { 4 }, \end{equation*}

\begin{equation} \label{a10} \sum _ { k = 1 } ^ { \infty } \left( \frac { ( 2 k + 1 ) ! } { k ! ( k + 1 ) ! } \right) ^ { 2 } \frac { 2 ^ { - 4 k } } { k } = \end{equation}

\begin{equation*} = 4 \operatorname { log } 2 + 2 - \frac { 4 } { \pi } ( 2 G + 1 ), \end{equation*}

and

\begin{equation} \label{a11} \sum _ { k = 1 } ^ { \infty } \frac { \zeta ( 2 k ) } { k ( 2 k + 1 ) 2 ^ { 4 k } } = \operatorname { log } ( \frac { \pi } { 2 } ) - 1 + \frac { 2 G } { \pi }, \end{equation}

where $\zeta ( s ) = \zeta ( s , 1 )$ denotes the familiar Riemann zeta-function.

References

  • [Fi] Steven R. Finch, "Mathematical constants", Encyclopedia of mathematics and its applications 94, Cambridge University Press (2003) ISBN 0-521-81805-2 Zbl 1054.00001
How to Cite This Entry:
Catalan constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Catalan_constant&oldid=38979
This article was adapted from an original article by Hari M. Srivastava (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article